[HN Gopher] Carrying Is a 2-Cocycle [pdf]
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       Carrying Is a 2-Cocycle [pdf]
        
       Author : mathgenius
       Score  : 24 points
       Date   : 2023-02-22 09:53 UTC (2 days ago)
        
 (HTM) web link (timothychow.net)
 (TXT) w3m dump (timothychow.net)
        
       | raphlinus wrote:
       | Carry is also a monoid. Consider addition of two binary numbers.
       | If the two digits are zero, then the output carry is zero no
       | matter the input carry. If the two digits are one, then the
       | output carry is one no matter the input carry. And if one is a
       | zero and the other a one, then the output carry is the input
       | carry. Call these values 0, 1, and X respectively.
       | 
       | Now consider a binary operator combining two such values. It is
       | associative and has X as an identity element, thus is a monoid:
       | \   0 1 X           +------         0 | 0 0 0         1 | 1 1 1
       | X | 0 1 X
       | 
       | Now you can express the carry output of bit i as the inclusive
       | scan (prefix sum) of these values from 0 to i. Because it's a
       | monoid, you can implement it efficiently in parallel. There's a
       | nontrivial literature in digital electronics exploiting this
       | basic fact.
       | 
       | Of course, to me everything is a monoid (including rendering
       | vector graphics paths), so it should come as no surprise that I
       | see things this way.
        
       | 082349872349872 wrote:
       | cf https://news.ycombinator.com/item?id=34884628
        
       | boxfire wrote:
       | Wish I saw this write-up in like 2009. The terminology in
       | Cohomology was pretty opaque to me until I (much later) learned
       | the concepts via a backwards mapping from learning deeper
       | applications of algebraic geometry. I would have learned that
       | much easier if I understood this easier.
        
         | 0xcafefood wrote:
         | Could you link to whatever source (paper, book, etc) you read
         | connecting deep learning to algebraic geometry? Sounds
         | interesting.
        
           | red_trumpet wrote:
           | I don't think "learning deeper applications of algebraic
           | geometry" has anything to do with "deep learning" in a
           | machine learning context.
           | 
           | My best guess is that OP had a course about rather abstract
           | homological algebra, which he only grokked after learning
           | about applications in algebraic geometry, which were "deeper"
           | in some sense.
        
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